We have sailed 3 Carnivals cruises all with our children. Our first year they were 8 and 10. They had a blast. It was because of their experience that we booked the 2nd and 3rd cruises.
Just last week we were at the Travel Agents doing a price comparison between them and Carnival. The prices were the exact same. Then I asked about the price diff between Carnival and RCCL. The price was in fact $200 per person difference.
I then asked about the activities for the kids (who are now 12 & 14). They suggested Carnival. Seems that RCCL has parents sign waivers, etc. and can not be so many feet away from the kids while they are participating in those wonderful activities (rock wall, water wave ride, and skating).
I think you (and the children) will be very happy with your choice.
Either way, safe sailing